Subgroup ($H$) information
| Description: | $A_4^3.C_4:S_4$ |
| Order: | \(165888\)\(\medspace = 2^{11} \cdot 3^{4} \) |
| Index: | $1$ |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$\langle(4,10,7)(5,8)(6,9)(13,16)(14,15), (2,11)(4,7)(5,8)(10,12), (3,6,9)(4,10,12) \!\cdots\! \rangle$
|
| Derived length: | $5$ |
The subgroup is the radical (hence characteristic, normal, and solvable), nonabelian, and a Hall subgroup. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $A_4^3.C_4:S_4$ |
| Order: | \(165888\)\(\medspace = 2^{11} \cdot 3^{4} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4^3.(S_3\times D_4)$, of order \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $S_4^3.(S_3\times D_4)$, of order \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| $W$ | $C_2\times A_4^3:S_4$, of order \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_2$ |
| Normalizer: | $A_4^3.C_4:S_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2\times A_4^3:S_4$ |